Even though room is square, with the items along the right side of room, looks like floor space would be a rectangle. I would use a rectangle shape placed as far over as you can so chair doesn’t potentially catch on its edge - suggest a flat weave as well jute will make chair’s mobility more difficult due to its thickness If you want a natural rug, sisal is typically thinner

I would measure from the door frame to the curtains. That will give you your diameter. Then, I'd measure exactly halfway along that measurement and put a piece of tape or other marking. Let's call that spot X. Then, from X, you can measure a half-diameter in any direction. That is the size of your rug. You probably do want to take a bit of string of the half-diameter length and draw a circle to see where the rug will be under your chair. If the walls behind the chair are not the same size, you probably want to use the shorter. For example, if one wall is 11 feet long and the other 10, you'd want to measure your diagonal from the edge of the 10 foot wall to 10 feet from the corner of the 11 foot wall. A circle enscribed in a square (ie the run inside the square made by the two walls behind the chair and 2 imaginary walls for the rest of the square) will have a diameter that is the diagonal measurement of the square (top right corner to bottom left corner). You can also do the whole thing with math with Pythagorus's theorem. (A^2 + B^2 = C^2) Measure the shorter wall, square that (A^2), double it (this is B^2, but A=B), then take the square root for your diameter. Pay attention to any obstacles, like baseboards, radiators or other things. You need to measure at the floor. If you want a bigger rug than the one that fits into your corner, you need to find a diagonal further into your room to measure, projecting from the walls. For example, a point midway into the door to an equal distance along the wall with the window. Since a circle enscribed in a square has the diameter of the diagonal of the square, as long as you have a square of floor space, you can figure out the diameter of the rug.

IMO I would do a cowhide with that chair and I have no idea what the tile means when see no round rug . BTW please find some art ,giant clocks are IMO never a decor item. Please post a couple more pics here if the whole space too. Post in a comment here do not start another post .

Rug should be the last decision (size/shape/style) based on you considering updating completely. Look for inspiration to help and consider budget to know if it’s possible. Even if updating down the road I’d skip a rug for now, as it won’t have a big influence on the space. If you could show your kitchen, you’ll get helpful advice for modern/minimalist if you are considering.